<<

CORE Metadata, citation and similar papers at core.ac.uk

Provided by Elsevier - Publisher Connector

JOURNAL OF ALGEBXA 28, 389-407 (1974)

The of a Morita Context

BRUNO J. MUELLER*

Department of , McMaster University, Hamilton, Ontario, Canada

Communicated by P. M. Cohn Received July 31, 1972

Every Morita context between rings R and S leads to an eq.uivalence between two quotient categories of the categories mod R and mod S. As consequences, one obtains a generalization of the Morita Theorems, and one constructs induced contexts between quotient rings of R and S. The concept of context-equivalence of rings is introduced and se&died. The last part reviews and reorganizes various topics utilizing the new notions and results.

1. PRELIMINARIES

DEFINITION 1.1. A (Morita) context consists of two rings R and S, two bimodules sPR and RQS, and two bimodule homomorphisms (called the pairings) (-, -): Q OS P -+ R and [-, -1: P OR Q + S satisfying the associativity conditions ~[p, 2’1 = (9, p) 4’ and p(q, p’) = [p, 41 p’. The inrages of the pairings are called the trace ideals of the context, and are denoted by TR and T, .

Remarks. This concept was introduced in [4], and under the name of preequivalence data in [5, II. 3.21 (cf. also [6]). It is extensively used in [I], and appears in disguise in many investigations. The most elegant (but apparently useless) description of a Morita context is to say that it is just an with two objects. Another description is obtained by constructing the fl = (: $) and its e = (i z): a context is just a ring together with an idempotent. We abbreviate a context by the symbol

* The author acknowledges support by the National Research Council of Canada, Grant A-4033. 389 CopyTight 0 1974 by Academic Press, Inc. All rights of reproducdon in any form reserved.

&T/28/7-2 390 BRUNO J. MiiLLER

[5, 1.4.21). The trace TR of the derived context of PE is also called the trace of PR , trace(P,). We need a somewhat more restricted notion of isomorphism of contexts than the one suggested by the description as additive categories, namely the following: Two contexts are isomorphic, (P, Q) g (P’, Q’>, if they involve the same pair of rings R and S, and if there exist bimodule iso- morphismsf: P + P’ and g: Q -+ Q2’ which are compatible with the pairings. A context will be called a subcontext of another one, (P, Q) < (P’, Q’), if they satisfy the same conditions but with f and g only monomorphisms. Thus, every subcontext is isomorphic to a context whose bimodules are actually submodules and whose pairings are obtained by restriction.

EXAMPLES 1.2. The only general method of constructing contexts is that of forming the derived context of a module; unfortunately it conceals the symmetry inherent in the original definition. A slight generalization restores this symmetry: From two modules X, and YA , define R = endo X, , S = endo YA , P = hom,(X, Y) and Q = hom,(Y, X), with pairings by composition. A context satisfies Ts = S iff PR and sQ are finitely generated projective, ,P and Qs are generators and the natural maps Q + P*, P -+ *Q, S + endo PR and S -+ endo aQ are bijective; iff PR is finitely generated projective, .P faithful and Q -+ P* surjective; iff ,P is a generator and Q- *P surjective. Then, TR is idempotent, TRQ = Q and PT, = P, and the context is isomorphic to (PR , CT) and (aQ, a), for the obvious choices of a (cf., [5, 11.3.41). A context satisfies T, = S and TR = R iff PR is a finitely generated projective generator and S-+endoP, is bijective (cf. [5,11.3.5; also Sect. 2.31). We finally mention the identity context (R, R> with pairings by multi- plication, and the trivial context with zero pairings.

1.3. This work was inspired by papers of Kato [26, 271 who proves our basic Theorem 3 in a slightly different language, for derived contexts. A great portion of our results was developed independently by Cunningham, Rutter and Turnidge [9] for the case of a derived context of a finitely generated . The vast majority of related papers are concerned explicitly or implicitly with the derived context of a finitely generated projective module PR [2, 3, 7, 8, 9, 12, 15, 26, 34, 35, 381 or a generator .P [23, 40, 421, the notable exception being [l]. The first case yields exactly the contexts (P, Q,> with Ts = S discussed before, and the second one the slightly more special contexts for which Pa satisfies in addition the bicom- mutator relation. One obtains a number of simplifications, mainly due to the equivalence hom,(P*, -) E P @a - and the idempotency of TR . THE QUOTIENT CATEGORY OF A MORITA CONTEXT 391

1.4. All rings are assumed to have identity elements, all modules are unitary, and mod R denotes the category of all right R-modules. X* is used for the dual hom,(X, R) of a right R-module X, , similarly *Y for a left R-module RY. dim X denotes the Goldie dimension of the module X, i.e., the largest number of nonzero summands in a direct sum of submodules. The symbols 5, 5, rad, ‘I), ‘$I and * will be used consistently for the torsion free class, torsion class, torsion radical, Gabriel filter (additive topology in [39, p. 12]), quotient category (Giraud , full subcategory of L&closed modules in [39, pp. 47-48]) and quotient functor of any hereditary torsion theory on a module category. Stenstriim [39] may serve as a general reference on hereditary torsion theories.

2. THE QUOTIENT CATEGORY

2.1. We start by collecting information about the hereditary torsion theory on mod R determined by an T of R.

PROPOSITION 1. For any two-sided ideal T of a ring R,

is the torsionfree class of a hereditary torsion theory. The corresponding Gabriel jilter is the set 3 of right ideals I such that ann,T = 0 implies annxr = 0 for all X E mod R, and the corresponding quotient category 58 consists of the X E mod R for which the natural X -+ hom,(T, X) is bijective. Proof. One verifies without difficulty that 5 is closed under products, submodules,extensions and essentialmonomorphisms, hence is a torsionfree class.Then I E D iff R/I is torsion, i.e., 0 = hom,(R/I, X) - ann,T for all. X E 5, providing the description of 3. The natural map

X 33~ti (t +-+xt) E hom,(T, X)

is injective iff ann,T = 0, i.e., XE 5. Under this assumption hom,(T, X) may be identified with the set of those elementse of the injective hull E of X which satisfy eT C X. Therefore the natural map is surjective iff eT C X implies e E X, i.e., annEfxT L- 0 or E/X E &. But for any hereditary torsion theory, X E 9l ifLFX, E/X E 3,

COROLLARY 2. The Gabriel Jilter consists of all right ideals containing T, a2 T = T2. Proof. Clearly T LL-:T” is necessaryfor the set of right ideals containing 392 BRCNQ J. Mi:LLER

T to be an idempotent filter. If this condition holds, consider the module X = R/(IT : T) for any IE 3. We have ann,T = 0 hence ann,I = 0, but iI = 0 hence icann,I = 0; i.e., 1 E(IT: T) or TCITCI. Remark. One would like to know which Grothendieck categories occur as quotient categories determined by ideals. The case of idempotent ideals is settled in [36].

‘I’HEORRM 3. Every Morita context betweenrings R and S indzcces an equivalencebetween the quotient categories2& and 21sof mod R and mod S determinedby the two trace ideals.

Proof. The bimodules sPR and RQ!S determine hom,(P, -): mod R + mod S and hom,(Q, -): mod S + mod R, and the pairings induce natural transformations

4: &ocm -j hom,(Q $31~ P, -) z homs(Q, hom,(P, -)> and ‘Pz i&-m -+ homR(P, hom3(Q, -I>, explicitely given by &(x)(q)(p) =: x(q, p). We claim $r to be isomorphic exactly if X E ‘LI, . Clearly ker & = ann,T where T = (Q, P) is the trace ideal; hence +x is injective iff X E 3. Under this assumption, any map 01E hom,(Q OS P, X) factors over ( , ),: Q OS I’ + T, since C (qi , pi) = 0 implies a(C 4i 0 A)(% PI, = aE 4i 0 Pi(%P,) = a(1 Qi0 [Pi Y4lP) = a(C4ii3i >41 OP) = “ICY& 1Pi)4 0 P) z:0 hence a(C qi @p,)T = 0 hence a(C qi 8~~) = 0. This produces an isomorphism

hom,( T, X) s ho&Q OS P, X) = hom,(Q, hom,(P, X)), whose composition with the natural map X -+ hom,(T, X) is just 4X . Therefore dx is bijective iff X -+ hom,(T, X) has this property, i.e., iff X E 5Z by the preceding proposition. 2.2. We sketch some results on isomorphisms of submodule lattices induced by a Morita context; the routine verifications are omitted. Results in the literature are special cases or follow immediately [12, 15, 23, 381. Consider first an arbitrary hereditary torsion theory on mod R, and the THE QUOTIENT CATEGORY OP A MORITA COKTEXT 393 lattice of submodules of a fixed R-module X. A submodule .A is called closed if XL4 E 8 (A E C,(X) in [39, p. 581). Each submodule A possesses a closure cl A, the intersection of all closed submodules containing A, or alternatively cl A = {x E X : (A : x) E a>. One obtains an A - H by cl A = cl H; cl A is the largest and the only closed member of the of A; A N B holds iff A and B have the same quotient module 2 = B C 2; and h induces a lattice homomorphism onto the lattice of all ‘?l- of X. The lattice of ‘VI-subobjects of X, the lattice of closed submodules of X, and the “factor lattice” of the lattice of all submodules of .X modulo m, are therefore isomorphic. If X is an S-R-bimodule, and if two hereditary torsion theories for S-left and R-right modules arc given, each subbimodule A possesses a biclosure bicl A, the intersection of all left and right closed subbimodules containing A, which determines another equivalence relation A m B; bicl A is again the largest and the only biclosed member in the class of A. Consider now a Morita context (P, Q2> and the quotient categories BF, and %, of Theorem 3. We say that X E mod A corresponds to YE mod S under (P; Q) if hom,(P, 2’) z Y as S-modules. Since in this situation the lattices of ‘LI-subobjects of X and Y are trivially isomorphic, all the lattices listed above are isomorphic, in particular:

PROIWSITIOS 4. If X, corresponds to Ys under a Morita cmtext (P, Q), then their lattices of closed submodules (with respect to the hereditary torsion theories deeternzined by the trace ideals) are isomorphic.

An example of corresponding modules are R, and Qzs ; indeed for all Y EL91, we have

hom,(hom,(P, A!), Y) G homs(homR(P, fi), hom,(P, hom,(Q, Y))

-.N_ hom,(&, hom,(Q, Y)) g hom,(Q, Yj s hom,&?, Y) since h is left adjoint to the inclusion functor, hence hom,(P, a) -z Q. In this case the isomorphism of the closed submodule lattices can be made quite explicite: A C R is mapped to B C Q iff R = - {q EQ : (4, P) C A) iR A -= {r E R : rQ C B). One derives that the two-sided ideals of R are mapped onto the subbimodules of Q, and that an ideal A of R is left and right closed iff its image 3 has the same property, all with respect to the hereditary torsion theories determined by the trace ideals. Composing with a similar isomorphism of the lattices of left and right closed subbimodules of Q and S, we obtain: 394 BRUNO J. MijLLER

PROPOSITION 5. For every Morita context between rings R and S, the lattices of left and right closed ideals (with respect to the hereditary torsion theories determined by the trace ideals) are isomorphic. Explicitly the left and right closed ideal A of R is mapped to (s E S : (Qs, P) C A).

If a hereditary torsion theory is determined by an idempotent ideal T = T2, then cl A = {x E X : XT C A) and the w-equivalence class of A contains a smallest member AT, which is the only T-accessible member of the class (a module i’k! is called T-accessible if MT = M (cf. [38])). In the bimodule situation if both hereditary torsion theories come from idempotent ideals, the biclosure is bicl A = {x E X : TxT C A) and the m-equivalence class has a smallest member TAT which is the only left and right T-accessible member.

PROPOSITION 6. Let both trace ideals of a Morita context (P, Q) between rings R and S be idempotent. If modules X, and Y, correspond to each other, then their lattices of TR- and T,-accessible submodules are isomorphic. The lattice of left and right T,-accessible ideals of R is isomorphic to the lattice of left and right T,-accessible ideals of S.

The assumption of the last proposition is for instance satisfied whenever one trace ideal is the whole ring, e.g., for the derived context of a generator or a finitely generated projective module.

DEFINITION 2.3. A Morita context is right normalized if the four natural maps P -+ Q*, Q -+ P*, R + endo Qs, S ---f endo PR are isomorphisms.

EXAMPLES. In 2.4 we shall construct right normalized contexts from an arbitrary Morita context. The derived context of a progenerator is both right and left normalized. The derived context of a generator PR is right normalized, but not left normalized unless PR is reflexive. A right normalized context is always derived from the pair (PR , 0) where u: S --+ endo PR is the natural map. The Morita Theorems [33, 4, 5, 311 establish a one-to-one correspondence between (1) the isomorphism types of category equivalences F: mod R -+ mod S, (2) the isomorphism types of Morita contexts (P, Q) between R and S satisfying TR = R and Ts = S, (3) the isomorphism types of bimodules sPR for which the natural map S --+ endo PR is isomorphic and PR is a progenerator. The correspondence is realized by associating with the bimodule sPR in (2) or (3) the functor hom,(P, -): mod R + mod S. The following generalization provides a classification of all category equiv- alences between full of module categories containing the rings. THE QCOTIENT CATEGORY OF A MORITA CONTEXT 395

THEOREM 7. Every additive category equivalence between full subcategories of mod R and mod S containing the modules Ii, and S, , has a unique maximal extension. and range of such maximal equivalences are quotient categories determined by ideals. Moreover there is a one-to-one correspondence between: (1) the isomorphism types of maximal category equivalences between fuh subcategories of mod R and mod S containing the modules R, and S, ; (2) the isomorphism types of right normalized Morita contexts between R and S; (3) the isomorphism types of bimodules sPR for which the natural maps S --z endo PR , R + hom,(T, R) and P -+ hom,(T, P) are bijective, where T -= trace(PR).

LEMMA 8. The derived context of PR is right normalized rj” the natural maps R -+ hom,(T, R) and P -+ hom,(T, P) are bijective, where T := trace(Ps).

Proof. By Proposition 1, the two natural maps are isomorphic iff R, P E 9X, the quotient category determined by T. By Theorem 3, this is the caseiff +s , & are isomorphic. Since we are considering a derived context, i.e., Q .-z hom,(P, R) and S = endo PR , we have

+R : A + hom,(Q, hom,(P, R)) = endo Qs and C& : P --f homS(Q, hom,(P, P)) = Q*,

hence $a and &, are isomorphic iff the context is right normalized.

Proof of the Theorem. Becauseof the lemma, the conditions on sPR in statement (3) may be replaced by the requirements that S -+ endo PR be bijective and that the derived context of PR be right normalized. We denote this modified statement by (3’) and establisha correspondencebetween (l), (2) and (3’). Theorem 3 provides a function I associatingwith any right normalized context (P, Q> the equivalence hom,(P, -): 21R-> +Lf, . We have R E 5XR since c$~: R s endo Qs s hom,(Q, hom,(P, R)) is isomorphic; similarly SE%&.. A function II associatingwith any equivalenceF: !93s+ ss, between full subcategoriesof mod R and mod S containing R, and S, , a bimodule sPR satisfying (3’), is obtained as follows: Let G: IcLf,-+ SR be an inverse of F; then F is representable by the bimodule P = GS, and G by FR G hom,(P, R) = P*. Therefore endo P,* = hom,(P*, hom,(P, R)) G 396 BRUNO J. MtiLLER

GFR E R, Pa* g hom,(P*, hom,(P, P)) z GFP e P, and endo PR G FP = FGS g S, by natural maps; i.e., the conditions of (3’) hold for P. A third function III associates with any bimodule satisfying (3’) the derived context of PR and U: S E endo PR , which is clearly right normalized. The compositions III II I and II I III are obviously identities (up to isomorphism). To investigate the relationship between F and I III II F, for any equivalence F: !B3, -+ Bs between full subcategories of mod R and mod S containing R, and S, , select an inverse G: BJ, -+ 23R of F and an isomorphic h: id+ GF. Then II F z GS, the bimodule representing F. Note first of all that if F’ is any extension of F and G’ an inverse ofF’, then G’ extends G hence G’S z GS, proving II F = II F’. III II F is the derived context of GS and u: S g endo GS, , and we observe GS* = hom,(GS, R) g FR, the bimodule which represents G. Therefore we obtain

R AR+ homs(GS*, hom,(GS, R)) s GFR L R, which is a bimodule isomorphism by naturality, hence given by multiplication with an invertible central element c E R. Naturality implies that $x is the composition of h,c with the isomorphism GFX ---f hom,(GS*, hom,(GS, X)) hence itself an isomorphism, for all X E 8, ; proving 23, C aIR . Therefore I III II F is the unique maximal extension of F, and the theorem is proved.

COROLLARY 9. If a right normalized context exists between rkgs R and S, then their centers aye isomorphic. Proof. The usual argument [4] establishes an isomorphism between the of R and the center of any full subcategory of mod R containing

R R’

OPEN PROBLEMS. Call an ideal T of a ring R admissible if it occurs as the trace ideal of a right normalized context. (1) For R = Z the ring of , T = Z is the only admissible ideal. Even if T = T2 and the natural map R + hom,(T, R) is isomorphic, T need not be admissible, a counterexample being T = @ Ki in R = l’J Ki , an infinite product of fields, Characterize the admissible ideals! (2) Every Grothendieck category is obtained as quotient category of a suitable mod R containing the module RR [17], but it is unlikely that each will occur as a quotient category determined by an admissible ideal. Characterize these Grothendieck categories! (3) Give more explicit descriptions of the modules PR of the third statement of the theorem, at least in special cases! THE QGOTIEKT CATEGORY OF A MORITA COSTEXT 397

2.4. We return to an arbitrary Morita context (P, Q) and the induced equivalence between the quotient categories SU, and SLT,determined by the two trace ideals, as described in Theorem 3. Let $a, denote an arbitrary quotient category of %IR, or equivalently a quotient category of mod R belonging to a Gabriel filter sR containing TR (cf. [16, p. 3691). The restriction of the original equivalence provides an equivalence between $3, and some quotient category ‘Bs of mod S belonging to some Gabriel filter $s containing T, . An explicit description of the relationship between the filters BR and ss is the following:

LEMMA IQ. ss consists of the right ideals L of S fog which P/I2, is $,-torsion. Proof. I, f as iff S/L is Bs-torsion, i.e.,

0 =- hom,(S/L, hom,(P, X)) e hom,(SL OS P, X) .g hom,(P/LP, X)

for all X E;‘Zr, . Let R 3 Y --+ P E & and S 3 s --t $ E S be the quotient rings of R and S in ‘&, and a, . Since ‘@, may be regardedas a full subcategoryof mod 8, we are in the situation. of Theorem 7, though one should note that the equivalence ‘&, -> a, need not be maximal if considered as an equivalence of sub- categories of mod I? and mod S. Therefore there exists a unique associated right normalized context between I? and S.

LEMMz4 1 I. The quotient module of PR is P = hom,(Q, s), with quotient map P 3 p + $ = [z] E P; similarly for Qs .

Proof. The quotient functor is left adjoint to the inclusion $3, C mod H, hom,(P, X) q hom,(hom,(P, hom,(Q, S)), hom,(P, X)) E hom,(S, hom,(P, X)) L%hom,(P, X) for all XE a, . The quotient map is obtained by choosing X = P and n taking the element corresponding to Ia, i.e., Y3(1) = I,[-, -1 -2 [-, -1. Remark. Explicitly the context between 8 and S is the derived context of the bimodule P, which arises as the image of S under the functor hom,(Q, -)I $2, and which represents the functor hom,(P, -)I ‘@R. Then hom,(P, a) E hom,(P, R) = & under the representing isomorphism f +> (p + > f( $)); therefore the context between I? and S is isomorphicx context p) and [ $, G]^ -= [p, q]. 398 BRUNO J. MtiLLER

3. NONDEGENERATE CONTEXTS

The considerations of this section are elementary, and most statements are easily verified. The main new concept is the equivalence relation for rings introduced in 3.2.

3.1. Associated with any Morita context (P, Q} are eight natural maps, e.g., P~pt+[p,-]EQ* and R3r+-+(q++rq)EendoQs.

DEFINITION. The context (P, Q) is called nondegenerate if all these natural maps are injective.

Remarks. In most of the individual statements to follow assuming nondegeneracy, this assumption can be weakened. It appears however that the full theory cannot be developed under a weaker hypothesis. A context (P, Q) is nondegenerate, iff the four modules ,P, PR , RQ, Qs and the two pairings are faithful (the latter meaning that (q, P) = 0 implies q = 0, and three analogous implications). The context derived from a module PR is nondegenerate iff PR is torsionless and faithful and the left annihilator of trace(P,) is zero. If (P, Q> is nondegenerate, then the Goldie dimensions of R, and QS coincide [l, Theorem 21. A context

PROPOSITION 12. A context is nondegeneratable ifs the two trace ideals are faithful as left and right modules (i.e., have zero left and right annihilators). In this case the submodules A, B are uniquely determined. -- Proof. If (P, Q> is a -- nondegeneration of (P, Q), then 0 = r(Q, P) implies 0 = (rQ, P) = (rQ, P) hence 0 = @ hence r = 0, i.e., the left annihilator of the trace (Q, P) is zero. Conversely if the four annihilators of the two traces are zero, we have [p, Q] = 0 iff (Q, p) = 0, since 0 = [p, Q] implies0 = Q[P,Qlb, QIP= (Q,P)(Q, PXQ, P) hence(Q, P) = 0. For A = {p E P : [p, Q] = 0} and B = (q E Q : (q, P) = 0} the induced pairings for (P/A, Q/B) are well defined, and the new context is nondegenerate. If (P/A’, Q/B’> is any nondegeneration of (P, Q), then p E A’ iff 5 = 0 iff 0 = [ 3, Q] = [p, Q], hence A’ is the submodule A used above, proving uniqueness.

Remark. Any nontrivial context between prime rings is nondegeneratable. THE QUOTIWT CATEGORY OF A MORITA CONTEXT 399

3.2. Given two contexts

(7 ):QCS:sf’+R, [ ,]:P.&Q-fS and

( ) ): vg,J), u-+s, [ ) ] : u5& v-.+ -4 the new pairings

define a new context between the rings A and fl, whose trace ideals are (Q, (K WY and W’, [P, Ql 0

LEMMA 13. If both (P, Q) and (U, V) are nondegenerate, then the new context is nondegeneratable.

PYOO~. The new trace ideals are faithful, indeed if 0 -=7 r(Q, (V,U)P) then 0 -= (YQ(V, U), P) hence 0 7: YQ( V, U) hence 0 = YQ hence 0 -; Y.

DEFINITION. For any two nondegenerate contexts (P, Q) and (ti, V) between rings R, S and S, A respectively, the (uniquely determined) nondegeneration of the new context between R, A will be called their nondegenerate composition and will be denoted by (P, Q) 0 (C, V). Nondegenerate composition is an associative (partial) operation. The identity contexts (R, R) act as identities. Existence of a nondegenerate context between two rings, is an equivalence relation, since nondegenerate composition establishes transitivity. We define formally:

DEFIW~IOK. Two rings R and S will be called context-equivalent, R w S, if there exists some nondegenerate context between them. Morita-equivalent rings arc clearly context-equivalent (cf. [41]). We list a few ring-theoretical properties, invariant under the new equivalence relation [I]: prime, semiprime, right primitive, Jacobson semisimple, no locally nilpotent ideals, no nil left ideals. A further invariant is the vanishing of the right singular ideal; more precisely we have the following result.

PROPOSITION 14. If the context (P, Q) is nondegenerate, and if one of Z(R,) 1:: 0, Z(P,) 7.: 0, %(QS) = 0, 2(&C;,) -= 0 holds, then all of them hold.

PYOO~. If Z(R,) =:- 0 and if pZ = 0 w h ere Z is a large right ideal of R, 400 BRUNO J. MCLLER then 0 == (Q,p)l hence (Q,p) := 0 hence p = 0, for any p E P; thus Z(P,) I= 0. Conversely if Z(P,) ~2 0 and rI = 0 for Y E R and large 1, then PrI z-z 0 hence PY = 0 hence Y =-10, thus Z(R,) =- 0. Now supposeZ(R,) --= 0 F:= Z(P,), and consider any s E Z(S,); then there is a large right ideal L of S with sL F- 0. Take any p E P and put I = {r E R : pr ELP}. This is a large right ideal of R; indeed if 0 =/ x E R then either px = 0 E LP hence x E I, or px # 0. In this casethere must exist q EQ with [px, q] + 0, and since L is large in S we have t E S with 0 + [px, q]t EL. Then there must exist p’ E P with [px, q] tp’ + 0, hence px(q, tp’) = [px, q] tp’ ELP hence 0 + x(q, tp’) ~1. But pICLP which impliesspI C sLP = 0, i.e., sp E Z(PR) =- 0. Sincep E P was arbitrary, SP = 0 hence s = 0 demonstrating Z(S,) L= 0. 3.3. Since the class of nondegenerate contexts between a fixed pair of rings may be quite large, we introduce an equivalence relation on it.

LEMMA 15. A subcontext of a nondegenerate context is nondegenerate ifit is nondegeneratable.

Proof. We may assumethat the modules U, V of the subcontext are actually subbimodulesof P, Q and that the pairings are obtained by restriction. If (ZI, U) : = 0 then (ZI,Q)( V, U) = (v, Q( V, U)) : = (v, [Q, V] U) C (v, U) :-= 0 hence (zi, Q) = 0 h ence v = 0 (cf. Sect. 1.I and the proof of Proposition 12).

DEFIKITIOS. Nondegenerate contexts between the same pair of rings are called equivalent, (P, Q) - (X, Y), if they are in the equivalence relation generated by the relation < (“subcontext”). This equivalence relation is compatible with nondegenerate composition, enabling one to talk about the composition of equivalence classes.Each equivalence classpossesses a composition-inverse, by the next result.

PROPOSITIOS 15. For any nondegenerate context (P, Q> o N (R, R). Proof. By definition of nondegenerate composition, (P, Q) o (Q, P) = QOsP> wh ere ~-Cqi@pi ~-0 iff 0 =(Cqi@pi,Q@P) := C (qi , [pi , QIP) = C (pi , PJ(Q, P) iff 0 -_ C (qi , pi>. ThereforeQ OS P may be identified with T -= (Q, P), and-~ this identification~ carriesthe pairings into multiplication. Consequently (Q OS P, Q 03) is a subcontext of , proving the proposition.

~OKOI.T.ARY 17. If four rings R, S, A, B aye all context-equivalent, then the classes of nondegenerate contexts between R and S, are in one-to-one correspondence with the classes of nondegenerate contexts between A and B. THE QUOTIEIST CATEGORY OF A MORITA COSTEXT 401

A more detailed study of the equivalence relation for nondegenerate contexts will be carried out in a continuation of this paper. We announce one result: Two equivalent nondegenerate contexts always have a common nondegenerate subcontext. 3.4. We return to the situation and notation of Section 2.4, for a non- degenerate context (P, Q).

LEMMA 18. If (P, Q> is a nondegenerate context, and if $a-rad R --.- 0, t&n $,-rad S = 0 and the induced context (f, &> is nondegenerate.

Proof. rad R -= 0 means R E $, and since PR is torsionless hence contained in a product of copies of R, WChave P E 3. Therefore if L E $uc hence P/LP $,-torsion, then hom,(P/LP, P) 7: 0. But for s E rad S there isL E ss with sL I= 0 hencesLP = 0, so s induces a map in hom,(P/LP, P), and consequently sP = 0 hence s = 0 since .P is faithful. Tf PE fi lies in the right annihilator of the trace of the induced context (P, @, then becauseeach a:E P” extends to an B E hom,(P, 8) s &, we have a(P)P C jz(p)P = (~9,f)V = 0 h ence Ta,P = 0 hence P -- 0, since R C I? is essential. Therefore the right annihilators of the traces of (P,&) are zero, and the nondegeneracy of the context follows readily.

Remark. We have not been able to decide whether is, without the assumption $,-rad R := 0. The largest hereditary torsion theory for which R is torsionfree, is called the Lambek torsion theory; the corresponding is the maximal (or complete) right quotient ring [39, p. 10; 31, Section 4.31). Nondegeneracyof a context (P, Q) implies annR 7; z 0 = arm, T, , i.e., R and S are torsionfree in the hereditary torsion theories determined by the trace ideals. Therefore the preceding considerationsapply to the Lambek quotient categories, and the last lemma and its symmetric analogueyield:

THEOREM 19. A nondegenerate context between rings R and S induces a nondegenerate and right normalized context between their maximal right quotient yings.

COROLLARY 20. The maximal right quotient rings of context-equivalent rings are context-equi7;alent.

4. APPLICATIOPS 4.1, Maximal Quotient Rings of I?ndomorphism Rings. Let (P, $3) be a nondegenerate context between rings R and S, and let (f, @ denote the induced context between their maximal right quotient rings P and .X. Then 402 BRUNO J. MiiLLER

Z g end0 Pp .= end0 PR , and one would like a more explicite description of the quotient module P of PR . This is possible at least in two cases: if P is nonsingular, f is the injective hull of P; if the Lambek torsion theory is perfect [39, p. 131, e.g., if P is semisimple artinian (cf. [37, 1.6]), then P=P@,P. A ring has a semisimple artinian maximal (classical) right quotient ring, iff it is right nonsingular, finite dimensional (and semiprime) (cf. [18,44, 371). Therefore ,Z’ is semisimple artinian (classical) iff P is nonsingular, finite dimensional (and S and/or R semiprime), cf. Section 3. If 2 has these properties and if dimQs < co, then P has the same properties.

PRO~OSITIOX 21. Let (P, Q) be a nondegeneratecontext betweenR and S, where S has a semisimple artinian maximal left and right quotient ring, and suppose dim Qs < co. Then R has a semisimple artinian maximal left and right quotient ring, and the inducedcontexts and

Symmetry implies equality, and consequently the map P -+ P of semisimple Z-modules is bijective. Similarly Q s Q; and Qz is a progenerator because it is finite dimensional and faithful over a semisimple . One obtains a Morita equivalence between Z and P g endo Qz g endo, P = endo= P, the maximal left quotient ring of R, and an isomorphism of contexts (P, Q) g

PROPOSITION 22. The following are equivalent for a ring S:

(1) S is context-equivalent to a right Ore domain, (2) S is prime, and its right Lambek quotient ring is isomorphic to the of a right , (3) S is a right nonsingular with a uniform right ideal. Remark. Faith calls such rings right Johnson yings, they may be charac- terized in several other ways [l 1, Sect. 14; 24; 29; 301. Proof. If S is context-equivalent to a right Ore domain R, then S is prime since R is prime (cf., Section 3.2). Any nondegenerate context between S and R induces a right normalized context (P, Q) between the right Utumi quotient rings 2 and A (a division ring), hence .Z G endo Pb is a full linear ring. If S satisfies (2) with right Utumi quotient ring ,Z, it is right nonsingular since 27 is regular; and eZ n S where e E 2 corresponds to the projection onto a one-dimensional subspace, is easily seen to be uniform. If (3) holds for S, consider the derived context of a uniform right ideal Qs . Since S is prime, left and right annihilators of trace (Q) are zero, hence the context is nondegenerate. R T- endo Qs lacks zero divisors (e.g., [43, Lemma 1.31) and dim& ..z dim Qs = 1, hence R is a right Ore domain

LEMMA 23. If R and S are two context-equivalent right Ore domains, then they are isomorphic to right orders in the same division ring A; moreover every nondegenmate context beta+een them is semi-isomorphic to a context consisting of subbimodules of A, z&th pairings by multiplication. Proof. Let (P, Q) be any given nondegenerate context between R and S. The induced context (P, Q) between the right Utumi quotient division rings A and A’ satisfies A’ s endo g endo A, z A since dim rj, = dim A’,! = 1; hence R and S are right orders in isomorphic division rings. Xote that P -+ P OR A s A is monomorphic since P is torsionless. Pick any 0 #pO~P; then SEI+S,EA where sp,@l =p0@6,, is a well-defined and injective ring homomorphism, identifying S with a certain subringofA.ThenP3pt-r&.EAwherep@l =po@6,,andQ3qi-> 404 BRUNO J. MtiLLER

(p, p,,) E R C A are injective bimodule homomorphisms, which carry the two pairings into multiplication.

COROLLARY 24. Right Ore domaim are context-equivalent, zy they are isomorphic to equivalent right orders in a division ying.

Proof. If R and S are context-equivalent, the lemma produces a non- degeneratecontext (P, Q) within a division ring A. Pick any nonzero p E P, q E Q; then qSp = (qS, p) C R and pRq C S. Since R was a right order in A, R and S are equivalent orders in A. Conversely if R and S are equivalent right orders in a division ring A, say xRy C S and ySx C R for nonzero elements x, y of A, put P = SxR and Q = RyS with pairings by multiplication; this yields a nondegenerate context between R and S.

COROLLARY 25. If two right Johnson rings are equivalent, then there is just one class of nondegenerate contexts between them.. Proof. It sufficesto prove the statement for right Ore domains. By the lemma, all nondegenerate contexts between such may be assumedto lie within a fixed division ring A. Let (P, Q) and (P’, Q’) be two such contexts, then (P n P’, Q n Q’) is a common subcontext, which is nondegenerate sincePnP’#OandQnQ’#O. 4.4. The Faith-Utumi Theorem. This section offers a new proof of this theorem, which though probably not shorter, might be more transparent than the original computational argument. It is basedon the following lemma, which is also the source of the density theorem in [I], and can be proved by a simple induction.

LEMMA 26 [I, Lemma 71. Let (P, Q) be a nondegenoate context over a right Ore domain R, and let U, C PR with dim U, = n < co. Then there exist elements u1 ,..., u, E P, q1 ,..., qnEQ and 0 #aER such that (9; ,uk) = ha.

THEOREM 27 (Faith-Utumi [13, 251). If S is a ying with classical right quotient ring A, , A a division ring, then there exists a right opdw A in A (not necessarily with identity element) such that S is isomorphic to an intermediate ring of A, C A, . Proqf. There is a nondegenerate context (P, Q) between S and some right Ore domain R, and dim PR = n < co. Select u1 ,..., u, E P satisfying the statement of the lemma, and put A = {a E R : there exist q1 ,..., qn EQ such that (qi , ule)= &a}; by the lemma A + 0. A is easily checked to be THE QUQTIENT CATEGORY OF A MORITA CONTEXT 405

a left ideal of R, and is therefore a right order in the quotient division ring A of R. Consider the induced context (f, Q) between the maximal right quotient rings A and .Ze A, . Since ur ,..., U, are R-independent, they form a A-basis for P = ,P 8s A. For any sequenceqr ,..., yn in the definition of A we have [z+ , ~$1E [P, Q] C S C 2, and since [u,. , q,](uk) = uj(qI , I+.) = Ci ui aij ?&,a, the endomorphism [z+, qa] is represented by the matrix (aij ?&z)~, with respect to the basis ur ,..., U, ~ Consequently A has the required properties. 3.5. Sk+ Rings. (cf. [IO, l&20,21, 281).If (P, Q) is any nondegenerate context between a S and an arbitrary ring R, then T, =-= S hence (cf. Sect. 1.2) PR and sQ are finitely generated projective, S z cndo PX z endo RQ and T == T”, where 1’ = T, . By Section 2.2 there is a lattice isomorphism between ideals of S and left and right T-accessible ideals of R, hence 0 and T are the only such ideals of R. Therefore T is contained in every nonzero ideal of R (since T is left and right faithful). Note also that R is left and right primitive (cf. 3.2). If R is also simple, PR becomesa generator hence S and R are Morita-equivalent. Conversely if a ring R possessesa smallest nonzero ideal T and a finitely generated projective module PR with trace(PR) = T, then the derived context of PX satisfies Ts == S and TR =: T :-= T2 and is nondegenerate; hence again by Section 2.2 and since 0 and T are the only left and right T-accessibleideals, S must be simple. Therefore:

~ROI~OSITIO~ 28. A ring R is context-equivalent to a simple ring S z$ A has a smallest nonzero ideal T and a Jinitely gerterated projective module PR with trace(PR) == T. Then for any nondegenerate context (P, Q> between R and S, PR and ,Q aYe Jinitely generated projective and S 2 endo PR 2 endo sQ- Context-equivalent siwaple Gngs are Morita-equivalent. In view of these results the following observation may be of interest:

PROPOSITIOx 29. 3’or a finitely generated projective module PR ovey a ring R with a smallest nonzem ideal T, the following ape equivalent: (1) trace(PR) = T,

(2) PR and 2X have the same simple factors, (3) all simple factors of PX aye faithfd, (4) P = PT.

Proof. Let trace(P) = T, and consider a simple factor sR/M of P, where M is a maximal right ideal. By projectivity, there is a map 01E P*

48NY3-3 406 BRUNO J. MijLLER

with P -+ RIM = P -+U R --+ RIM, and since a(P) C T but a(P) @ M, we have T $ M hence RIM G TIT n M. Given on the other hand a simple factor T/K of T, there must be p E P* with p(P) @ K hence T/Kg p(P)//l(P) n K is a factor of P. Nonzero factors of T are clearly faithful. If P # PT, there is a maximal PT C NC P since P is finitely generated, hence (P/N)T = 0; contradiction to (3). That PT = P implies trace(P) = T, is equally obvious.

REFERENCES

1. S. A. AMITSUR, Rings of and Morita contexts, J. Algebra 17 (1971), 273-298. 2. F. W. ANDERSON, Endomorphism rings of projective modules, Math. 2. 111 (1969), 322-332. 3. M. AUSLANDER AND 0. GOLDMAN, Maximal orders, Trans. Amer. Math. Sot. 97 (1960), l-24. 4. H. BASS, The Morita theorems, mimeographed notes, 1962. 5. H. BASS, “Algebraic K-Theory,” Benjamin, New York, 1968. 6. P. M. COHN, Morita Equivalence and Duality, Queen Mary College Math. Notes, 1966. 7. R. S. CUNNINGHAM, Morita equivalent rings of quotients, Notices Amer. Math. Sot. 19 (1972), A-73. 8. R. S. CUNNINGHAM AND E. A. RUTTER, On perfect injectors and perfect projectors, Notices Amer. Math. Sot. 18 (1971), 114. 9. R. S. CUNNINGHAM, E. A. RUTTER, AND D. R. TLJRNIDGE, Rings of quotients of endomorphism rings of projective modules Pacific J. Math. 41 (1972), 647-668. 10. C. FAITH, Simple noetherian rings, Bull. Amer. Math. Sot. 70 (1964), 730-731. 11. C. FAITH, “Lectures on Injective Modules and Quotient Rings,” Springer Lecture Notes in Math., 49, 1967. 12. C. FAITH, A correspondence theorem for projective modules and the structure of simple noetherian rings, Bull. Amer. Math. Sot. 77 (1971), 338-342. 13. C. FAITH AM) Y. UTUMI, On noetherian prime rings, Trans. Amer. Math. Sot. 114 (1965), 53-60. 14. E. H. FELLER AND E. W. SWOKOWSKI, The ring of endomorphisms of a torsionfree module, J. London Math. Sot. 39 (1964), 41-42. 15. T. V. FOSSUM, Lattice isomorphisms between Morita related modules, Notices Amer. Math. Sot. 18 (1971), 361. 16. P. GABRIEL, Des categories abeliennes, Bull. Sot. Math. France 90 (1962), 323-448. 17. P. GABRIEL AM, N. POPESCO, Caracttrisation des categories abeliennes avec gtnerateurs et limites inductives exactes, C. R. Acad. Sci. Paris 258 (1964), 4188-4190. 18. A. W. GOLDIE, Semiprime rings with maximum condition, Proc. London Math. Sot. 10 (1960), 201-220. 19. R. HART, Endomorphisms of modules over semiprime rings, J. Algebra 4 (1966), 46-51. 20. R. HART, Simple rings with uniform right ideals, J. London Math. Sot. 42 (1967), 614-617. TIiE QUOTIENT CATEGORY OF A MORITA CONTEX’I 407

21. H. HAIW ANU J. C. ROISOS, Simple rings and rings A4orita equiralent to Ore domains, Proc. London j%lath. Sot. 21 (1970), 232-242. 22. N. JACOBSON, Structure of Rings, Amer. Math. Sot. Publ. 37, 1964. 23. A. \‘. JATEGAOPX~I~, Endomorphism rings of torsionless modules, Trans. Amer. LMath. Sot. 161 (1971), 457-466. 24. K. E:. JOHXSOZI, Quotient rings of rings xvith zero singular ideal, P~ci’ic 1. Math. 11 (1961), 138551392. 25. R. E. JOIISSON, Prime matrix rings, Proc. Anrrr. :Vlath. Sot. 16 (l965), 1099-1105. 26. 1‘. KA,I.o, Dominant modules, /. Algebra 14 (1970), 341-349. 27. ‘I’. K~ro, U-distinguished modules, /. Algebra 25 (1973), 15-24. 28. K. KOH, On simple rings with maximal annihilator right ideals, Ca~cxf. ,Math. 1Mf. 8 (1965), 667-668. 29. K. KOH AND A. C. ~IEWBOXX, Prime rings with maximal annihilator and maximal complement right ideals, Proc. Amer. Math. Sot. 16 (1965), 1073-1076. 30. J. IAMIEK, On the structure of semiprimc rings and thsir rings of quotients, Canad. 1. Muth. I3 (1961), 392-417. 31. J. 12AMBEK, “Lectures on Rings and Modules,” Blaisdell, Waltham, kIass., 1966. 32. R. W. IbfILLER, Quasi-generators, preprint. 33. K. MorIOHITA AND T. TACHIKAWA, QF-3 rings, preprint. 36. J. E. ROOS. CaractPrisation des categories qui sont quotients de categories de modules par des sous-categories bilocalisantes, C. H. Acad. Sci. Paris 261 (I 965), 4954-4957. 37. I;. I,. SASDOWERSKI, Semisimple maximal quotient rings, Truns. .4mer. ,Wath. Sac. 128 (1967), 112-120. 38. 1:. L. SASDOMERSKI, Modules over the cndomorphism ring of a finitely generated projective module, PYOC. Amer. Math. Sot. 31 (1972), 27-31. 39. B. STESSTR~~M, “Rings and Modules of Quotients,” Springer Lecture Notes in >Iath. 237, 1971. 40. II. ~~‘AcHIK(AwA, On splitting of module categories, Math. Z. 1 II (1969), 145---l 50. 41. II. R. 'I'CRNIDGE, Rings of quotients of Morita equivalent rings, AYotices Ameu. 34ath. Sot. 16 (1969), 114. 42. 1:. R. \VILL~WI), Properties of projective generators, Moth. Ann. 158 (1965), 3 52-3 64. 43. J. %‘I. ZEL~IAxOWIC?, Endomorphism rings of torsionless modules, J. .4lgebra 5 (1967), 325-341. 44. J. Xl. ZEL~l.4NOWIW, A shorter proof of Goldic’s theorem, Cawxl. IMath. BuU. 12 ( l969), 597-602. 45. J. 11. ZELMANOWIT.~, Semiprime modules with maximum condition, J. Algebra 25 (1973), 554-574.